Electromagnetic Induction: The creation of an electric current by a changing magnetic field.

Transcription

1 Inducion 1. Inducion 1. Observaions 2. Flux 1. Inducion Elecromagneic Inducion: The creaion of an elecric curren by a changing magneic field. M. Faraday was he firs o really invesigae his phenomenon o is fulles Fig. 1 Observaions magne N If a permanen magne is insered ino a curren loop, a curren is deeced while he magne is moving. The curren is only presen while he magne is moving. S coil ammeer More Observaions baery swich coil 1 If wo coils where near each oher, and curren allowed o pass hrough coil 1, a he momen he curren begins o flow hrough coil 1, a curren is also measured in coil 2. coil 2 ammeer updaed on J. Hedberg 218 Page 1

2 One More Observaions N Lasly, if a coil was moved nearby a permanen magne, a curren was deeced during he moion. S The power which elecriciy of ension possesses of causing an opposie elecrical sae in is viciniy has been expressed by he general erm Inducion; which as i has been received ino scienific language, may also, wih propriey, be used in he same general sense o express he power which elecrical currens may possess of inducing an paricular sae upon maer in heir immediae neighborhood, oherwise indifferen. I is wih his meaning ha I purpose using i in he presen paper. M. Faraday, Flux z air flow Las ime we saw flux was when discussing he elecric field. We used he analogy of caching wind in a hoop. y x z air flow y x updaed on J. Hedberg 218 Page 2

3 Magneic Flux We can apply he same formulaion o magneic fields. We can imagine loops ha migh have differen areas, and differen orienaions wih respec o he magneic fields. The magneic flux will be defined in a very similar way. Φ = da For one loop, in a uniform field, he flux will be given by he area of he loop and he magneic field srengh and he cosine of he angle beween he loop normal and he field. Φ = Acos θ [Flux SI uni is called he 2 weber, 1 Wb = T m ] Quick Quesion 1 A curren flows hrough he gold wire as shown. Three idenical rings are locaed a equal disances away from he cener of he wire. Which of he hree rings shown will have he greaes amoun of magneic flux? (D if all he same) updaed on J. Hedberg 218 Page 3

4 Example Problem #1: A circular loop surface normal makes an angle of 3 degrees o he horizonal. A magneic field, parallel o he horizonal axis peneraes he loop and has a srengh of.5 T. If he loop is 1 cm in diameer, wha is he magneic flux hrough he loop. Example Problem #2: The long wire carries a curren of 1 A. Wha is he magneic flux hrough he loop? Lenz' Law 1. There is an induced curren in a conducing loop if and only if he magneic flux hrough he loop is changing. 2. The direcion of he induced curren will be such ha is induced magneic field will oppose he change in flux. Faraday's Law The magniude of he emf Einduced in a conducing loop is equal o he rae a which he magneic flux hrough ha loop changes wih ime. Φ E = dφ d updaed on J. Hedberg 218 Page 4

5 How o change he flux Change he posiion of he loop wih respec o he magneic field Change he srengh of he magneic field Change he angle beween he loop and he magneic field. updaed on J. Hedberg 218 Page 5

6 Quick Quesion 2 A ring is locaed in a region wih a magneic field as shown. The srengh and direcion of he magneic field are ploed as a funcion of ime. ank he magniude of he emf ( E ) of his saionary loop during he following ime inervals. 1. a > b > c > d > e 2. a > b > c > d > e 3. e > d > c > b > a 4. b > d = e > a = c 5. a = b = c = d = e a b c d e Quick Quesion 3 A The hand crank is urned Which plo shows he flux hrough he loop as a funcion of ime? C D E F A field increasing in he negaive Z direcion will induce a curren ha creaes a field in he posiive he Z direcion. These wo fields will cancel such ha he change in flux hrough he ring is zero. updaed on J. Hedberg 218 Page 6

7 In his case, he field begins direced in he negaive direcion. I hen begins decreasing in magniude as a funcion of ime. Now we increase in he posiive direcion. This will induce a flux canceling field in he negaive direcion. Quick Quesion 4 The meal loop is locaed in a uniform, bu seadily decreasing magneic field. Wha is happening in he loop? 1. A curren is induced in he clockwise direcion. 2. A curren is induced in he couner-clockwise direcion. 3. No curren is induced in he ring. 4. Can' say - we need o know more abou he siuaion. updaed on J. Hedberg 218 Page 7

8 curren Quick Quesion 5 velociy The meal loop on he righ is moving owards he curren carrying wire a a uniform velociy. Wha is happening in he loop? 1. A curren is induced in he clockwise direcion. 2. A curren is induced in he couner-clockwise direcion. 3. No curren is induced in he ring. 4. Can' say - we need o know more abou he siuaion. Quick Quesion 6 ring A meal loop is inside a solenoid wih a curren passing as shown. If he meal loop is released from res inside he solenoid and allowed o fall down he cenral axis under he influence of graviy, wha will happen in he loop? curren 1. The induced curren will increase as he ring acceleraes down. 2. There will be no induced curren as long as he ring is in he solenoid. 3. There will be an induced curren ha is consan. updaed on J. Hedberg 218 Page 8

9 Quick Quesion 7 Consider he siuaion shown in he drawing. A conducing loop is conneced o a resisor. The resisor and loop are a res in a magneic field ha is direced oward you. Wihin a shor period of ime he magneic field is reduced o one half of is iniial value. Which one of he following saemens concerning an induced curren, if any, in he loop is rue? 1. During he ime he magneic field is decreasing, a curren is induced ha is direced counerclockwise around he loop. 2. During he ime he magneic field is decreasing, a curren is induced ha is direced clockwise around he loop. 3. No curren is induced in he loop a any ime. 4. A curren is induced ha is direced clockwise around he loop, which also coninues afer he magneic field aains a consan value. 5. A curren is induced ha is direced counerclockwise around he loop, which also coninues afer he magneic field aains a consan value. Example Problem #3: The solenoid shown has 1 urns of wire. ased on he curren plo for he large solenoid, wha is he curren in he small loop (resisance of.1 Ω) as a funcion of ime? Plo i. updaed on J. Hedberg 218 Page 9

10 Quick Quesion 8 v As he gray bar is pulled o he righ, wha will happen in he circui? 1. A curren will flow in he clockwise direcion. 2. No curren will flow. 3. A curren will flow in he couner-clockwise direcion. Example Problem #4: The blue square sars moving o he righ a.5 m/s Wrie a piece-wise equaion ha you could use o predic he overlap (purple) area as a funcion of ime. Quick Quesion 9 A Assuming he loop ravels a a uniform velociy, which plo could represen he induced currens in he loop as he loop moves ino and hen ou of he magneic field region. v C D E updaed on J. Hedberg 218 Page 1

11 x L v Since P = Fv, we can use his o figure ou he rae a which work is done when pulling a loop hrough a magneic field. The magniude of he flux is equal o Φ = A = Lx b The emf is hen given by: E = dφ d = Lv Thus, he curren hrough he loop will be equal o: I = Lv x L v Using he force law: F = IL, we can deermine he force on he lef mos secion of he wire. F = IL = 2 L 2 v This can be used o find he power needed: b P = Fv = 2 L 2 v 2 x Or, we could use: P = I 2 o figure ou he hermal energy creaed by he circui. L v Lv P = ( ) = 2 2 L 2 v 2 which comes ou o be he same hing. b updaed on J. Hedberg 218 Page 11

12 Eddy Currens Eddy Currens are formed in conducors as hey move in a magneic field. They can be used in braking sysems for less wear. Quick Quesion v v Here are wo scenarios. On he op a wire wih side lengh 8 unis is being pulled ou he magneic field a speed v. On he boom are 4 wires each of side lengh 2 unis, also being pulled ou of he field a he same speed. Which siuaion will experience a larger resisive force? 1. The upper wire 2. They will boh exprience he same force 3. The lower group of wires updaed on J. Hedberg 218 Page 12

13 Quick Quesion 11 If he copper ring is le go from res above his bar magne, wha will happen? 1. The ring will fall like a freely falling objec (i.e. a z = g = 9.8m/ s 2 ) 2. The ring will quickly reach a erminal velociy and falling slower han if here was no magne. 3. The ring will accelerae down a a rae faser han g. (i.e. a z > 9.8m/ s 2 ) 4. The ring will fall a lile bi, bu hen say hovering a disance above he magne. Cu Tube Magne S S S N induced field N N increasing flux induced curren ries o cancel change in flux magne s field r If we sar wih a uniform field filling an cylindrical area, and pu a meal ring of radius r inside he area. We can predic he induced curren in he ring if he field were o sar increasing. E = dφ d updaed on J. Hedberg 218 Page 13

14 We should also be hinking abou wha is needed o creae a curren. Our wo crieria from before were: r 1. A place o go (i.e. a conducion pah) 2. A reason o go here. (i.e. an elecric field) Quick Quesion 12 If here is no meal ring, will here be an elecric field if d/d? 1. No, no charges mean no elecric fields 2. Yes, here jus has o be 3. Maybe Indeed, if here is a changing field, hen here will be an elecric field creaed. Elecric Field Lines In words: "A changing magneic field creaes an elecric field" In mah: E ds = dφ d (Jus hink abou wha emf was) This is Faraday's Law in anoher form updaed on J. Hedberg 218 Page 14

15 Example Problem #5: Find he elecric field as a funcion of posiion. Elecric Field Lines Inducors The solenoid has many useful applicaions. (Car saring, speakers, swiches, ec) We'll ake a look now how i's used in elecronic circuis. We'll sar by adding a new device o our lis of componens: he inducor. Inducance We can quanify he effec of hese devices by describing heir inducance, L: where, N is he number of windings, Φ is he magneic flux, and I he curren passing hrough. The SI uni of inducance is he Henry ( T L = NΦ I m 2 /A) updaed on J. Hedberg 218 Page 15

16 Since for a solenoid, he magneic field inside is given by: = In μ and, he quaniy N Φ = nla, we can wrie for he inducance: NΦ L = (nl)( μ In)A = = μ n 2 la I I L l = μ n 2 A Jus like capaciance, his is only dependen on he geomery of he device. We could also wrie he inducance as: L = μ N 2 A l Self-Inducion If here is a changing curren in he large solenoid, hen here will be an induced curren in he ring. There is however, an induced emf E in he original, large solenoid oo. This process is called Self-inducion updaed on J. Hedberg 218 Page 16

17 Self-Inducion Self inducion follows Faraday's Law jus he same. For any inducor: N Φ = LI Using Faraday's law: E L = dnφ d Which can be combined o give: E L di = L d Thus, he magniude of he self-induced emf is given by he rae of change of he curren, and no he curren iself. increasing I The curren I is increasing in he op figure. The self-induced emf is shown in he direcion such ha i opposes he increase. E L In he lower circui, he currren I is decreasing, and he selfinduced emf appears in a direcion such ha i opposes he decrease. The amoun of emf induced is given by he dervaive of he curren, and he inducance L of he inducor. di d decreasing I L circuis a b S L We can pu inducors in our familiar circuis and see he effecs. Here is an inducor L in series wih a resisance. We already know he inducor will oppose a change in curren. updaed on J. Hedberg 218 Page 17

18 asic Circui Our basic circui behaved like his afer he swich was urned on. The curren insananeously rose o he value given by I = V / However, in naure, nohing happens ruley insananeously. If we zoom in on ha sep, we'd see i's acually a smooh curve. L circui a b S L Since he inducor resiss he change in flux (or curren), hen we would expec ha by including one in his simple circui, we would see a change in he slope of he curren hrough he circui. L circui - Loop Analysis a b S L 1. The resisor: The poenial drop across he resisor is given by Ohm's law: -I 2. The inducor: Since he curren is changing (from zero o no zero), here will be an induced emf in he inducor. The magniude is given by: L di. Since he d curren is geing larger, we expec his emf o be direced agains he flow of curren, hus he sign di should be negaive: E L = L d 3. Lasly, he baery: The baery has is own emf, E All ogeher, we find he sum o be: di I L + E = d from he loop law. updaed on J. Hedberg 218 Page 18

19 a b S And so we end up wih: L di L + I = E d This is a differenial equaion for I(). One could find a soluion using sandard mehods and would arrive a: E I = (1 e /L ) or, by defining a 'ime consan' τ = L/ E I = (1 e /τ ) V V V (resisor) Afer he swich closes, he volage across he resisor will increase unil i reaches E, while he volage across he inducor will decrease from E o zero. The ime consan τ deermines how quickly hese changes occur. V (inducor) A similar analysis will ell us wha happens when we disconnec he circui from he baery and le he curren decay. Energy Sored in a magneic field When we separaed wo opposie elecric charges, his acion required work. This energy we invesed in he process was hen sored in he elecric field. In a similar way, we can alk abou he energy sored in a magneic field. updaed on J. Hedberg 218 Page 19

20 Energy We sar wih he loop rule for he simple L circui: Muliply boh sides by I yields: di E = L + I d EI = LI di d 1. The lef erm is he familiar power of he baery. (The poenial imes he rae of charge flow) Or, he rae a which he emf device delivers energy o he circui 2. The erm I 2 is he energy dissipaed by he resisor in he form of hermal energy (hea) 3. Since energy is conserved, he amoun ha doesn' go ino hea, mus herefore be pu ino he magenic field of he inducor. This erm accouns for ha amoun of energy. + I 2 du d = LI di d or d U = LIdI This is somehing we can inegrae o ge he oal energy. Inegraing yields d = LIdI Which is he energy sored in an inducor, or he magneic energy. U U I 1 U = L 2 I 2 updaed on J. Hedberg 218 Page 2

21 Example Problem #6: This elecromagne has an inducance of 76 Henries. Is field o curren raio is.157 Teslas per Amp. If i's operaing a 16 Tesla, wha's he poenial energy sored in he magne. Compare his o somehing more commonplace. updaed on J. Hedberg 218 Page 21